The myth about traditional math education

Garelick looked at math books and methods used in the ’40s, ’50s and ’60s.

Mathematical algorithms and procedures were not taught in isolation in a rote manner as is frequently alleged. Concepts and understanding were an important part of the texts.

Then and now, nobody argues for memorization without understanding, he adds.

Traditional math education was working reasonably well, Garelick argues. In Iowa, test scores rose steadily until about 1965, and then declined dramatically for a decade. This pattern was repeated in Minnesota and Indiana.

Source: Congressional Budget Office (1986)

Some researchers blame increased drug use and the rise in divorce and single-parent families for the decline. Garelick blames progressive education which called for student-centered, needs-based courses.

After taking not-so-early retirement, Garelick is now a student math teacher at a California junior high school.

Comments

Well, the math books I had growing up were all rote memorization and plug-n-chug. At the top of the page there were a bunch of examples of how to solve a particular type of math problem with no explanation of the underlying concepts. Then we’d be assigned a long list of practice problems. Lather, rinse, repeat. I was good at memorization so I got all A’s in math and 700 on my SAT-M, but I didn’t have a clue as to why the algorithms worked.

If you had asked me a few years ago to solve a division by fractions problem, I could’ve quickly calculated the correct answer. But if you had asked me why I inverted and multiplied, I couldn’t have given an answer beyond “because that’s the way you do it.” Now, after teaching my kids Singapore Math, I actually understand the reasons behind the algorithm and could give a conceptual explanation.

Some texts were better than others; it depends on the era. The text I reproduced in my article was from 1948 and the explanation of fractions was anything but rote, and offered a good explanation and shows why fractions are the same as division: a concept that is brought out in Singapore’s books.

While you now can give reasons behind the algorithms and conceptual explanations based on your reading of Singapore Math, you are an adult. I have taught students with Singapore Math. Some of them understand the concepts behind the procedures–some don’t. But all can do the procedures and problems. If the training you had as a child enabled you to score 700 on your SAT, then you likely were able to solve many problems. Would understanding why the long division and “invert and multiply” rule have enhanced your problem solving ability?

I think understanding the why’s behind the algorithms helps students who aren’t as good as I was at memorizing. A student who understands the reason why he/she places a 0 in the ones place of the 2nd partial product when doing a double-digit multiplication is less likely to forget to put it there.

I agree that traditional math was probably not as good as Singapore on concepts, but it was far better than the various spiral curricula kids are getting today, which doesn’t seem to teach them even everyday kinds of skills that used to be taken for granted. Even in my small-town school, kids (very few went to college) did learn math facts, the standard algorithms and how to apply them, how to manipulate fractions and the relationships among fractions, decimals and percentages. Calculating a tip or sales tax, balancing a checkbook, and calculating interest were things most people could do – and that was before calculators!

When my son was in preschool eleven years ago, I remember thinking about all of the things I didn’t like about the “traditional” math I had when I was growing up. One might claim that the emphasis was all on mastery of algorithms, but it’s not that simple. Proper mastery of skills does not preclude understanding. Textbooks didn’t offer just rote learning. However, I would have preferred a more rigorous presentation of the larger mathematical context. Does the math we have now do that? No. It flips the presentation completely around from a bottom-up, skills approach to a top-down understanding approach. But what is this understanding and critical thinking they talk about? It’s pictures of pieces of pie. It’s a very basic conceptual understanding that leaves students completely helpless when are confronted with things like adding rational expressions.

They claim balance and that they work down from the top to achieve mastery of skills, but it doesn’t happen. Students end up with vague conceptual understandings and mastery of very little. I would wager that those old, so-called, rote students had more mathematical understanding than modern Everyday Math kids. Skills without understanding you can fix. Understanding without skills is meaningless. We don’t have better math curricula now. No matter what the issues were with traditional math, what we have now has gone in the wrong direction. They are trying to make less sound like more.

Most people with technical backgrounds don’t want to go back to the traditional math days, but educators can’t seem to resist the argument. When the discussion focuses on something like Singapore Math versus Everyday Math, the real differences show up. It’s really about high versus low expectations; math for future scientists versus math for the rest. Unfortunately, the new, low expectation math guarantees that kids will never get there without outside help. I managed to get to calculus in high school without any help at home. That would never happen nowadays. I had to help my son every step along the way, but his schools will gladly accept credit for his success.

The rote argunment is just a way to focus attention away from all of the other pedagogical and curriculum changes they made. If they argue the generalities enough, then nobody will pay attention to the details.

I’m a science teacher, and I have to basically teach the math concepts the kids don’t understand. I hate it. It takes time from teaching my subject area. I should NOT be getting kids in high school that can’t see a fraction, and understand that if they divide the numerator by the denominator, they will get the equivalent fraction. And that fractions and decimals are just different ways of expressing the same number.

So, until the elementary and middle schools solve THAT problem, they haven’t succeeded. ALL kids who will be getting a high school diploma should have at least that level of math competency by the end of 8th grade. To send them to high school without that level of skill is virtually guaranteeing that they will fail. But, instead of putting the blame on the elementary or middle schools, it then becomes the high school’s “fault”.

A new curriculum may yield a short-term gain because it’s evaluated by true believers, the scientists said.

“Novelty-based enthusiasm, sample bias, and anecdotes account for much of the glowing characterization of [single-sex] education in the media. Without blind assessment, randomized assignment to treatment or control experiences, and consideration of selection factors, judging the effectiveness of innovations is impossible.”

Change the words she added in the brackets and it applies perfectly to math instruction.

In fact, you can apply those words to just about everything in education. Changes may work, they may not, but we just never know. So we fight with anecdotes and ideology.

I teach 6th grade math and intervention currently and believe that there are certain foundational skills that kids do need to master – period. If a kid can’t multiply whole numbers or add two integers they won’t be successful in pre-algebra let alone algebra. SteveH above said “Understanding without skills is meaningless” – I agree!
Also, what we used to teach in 6th grade 20 years ago is what is being introduced/taught as early as 3rd grade in some states. Interesting comments….thanks!

Graduation from 8th grade should be conditional upon reaching a minimum level of mastery of basic English Language and Math skills (modified for ESOL students, of course). Sending them to high school is cruel if there isn’t a hope in H–L that they will graduate.

If they don’t pass the test, there are several choices:
- high school prep – NOT in the high school, but another facility. They work to get the skills, before they get to enroll in high school. This extends their time, but if they get up to skill level, they can still be entitled to 4 years of high school.
- frank discussion with parents. They need to agree that their kid will NOT get a diploma, but a certificate of attendance, and a marketable skill – what we used to call a trade diploma.
- intensive summer school, AND mandatory tutoring after school/Saturday school. Failure to report get them put in one of the other options.

I say this because, unless a kid is a fairly low IQ, they can achieve that skill level. They may have been badly taught. They made have had poor attendance. They may have goofed off, and played around, and been passed. They may have moved around a lot.

Lots of reasons for not having reached the skill level. NO good reason for schools not taking the responsibility for helping that kid reach that skill level. Start with the belief that it’s possible. Kids can learn – if the motivation is strong enough.